Finite Group Databasebeta

← #45663486

Subgroups — #45663486

Order 32896.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30657402C211yesmaximal, characteristic
2#30657089C211yesmaximal, characteristic, complemented
3#30657033C211yesmaximal, characteristic, complemented
4#30657028C211yesmaximal, characteristic
5#30656467C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic
7#332×C257C211yesmaximal, characteristic, complemented
8C2×Q8×D257—21no—
9#22551592C2211yescharacteristic
10#22551591—21no—
11#22551570—22no—
12#22551568—21no—
13C4×Dic514C2211yescharacteristic
14C4×D1028C2211yescharacteristic
15#22551554—21no—
16#22551526—21no—
17#22551524—21no—
18#22551517—21no—
19#22551515—22no—
20#22551514—21no—
21C4×#14523339C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514—22no—
26C2×Dic514—21no—
27C2×Dic514D411yescharacteristic
28Q8×D257—42no—
29Q8×D257—42no—
30C2×D1028D411yescharacteristic
31C2×C4×D257—22no—
32C2×C4×D257—21no—
33C2×#14523339—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257D411yescharacteristic
37C4⋊Dic257D411yescharacteristic
38#16861597—22no—
39#16861597—21no—
40#16861595—22no—
41#16861595—21no—
42C4×Dic257—21no—
43C2×Q8×C257—21no—
44C2×C8×C257—21no—
45C2×C8×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257D412yes—
48C42×C257C2311yescharacteristic
49C22×D257—21no—
50Dic514—21no—
51Dic514—44no—
52Dic514—42no—
53Dic514SD1612yes—
54D1028—21no—
55D1028Q1612yes—
56C2×Dic257—21no—
57C2×Dic257—21no—
58C2×Dic257—21no—
59C2×Dic257—41no—
60C4×D257—44no—
61C4×D257—41no—
62C4×D257—41no—
63#14523339—41no—
64Q8×C257—41no—
65Q8×C257—41no—
66C2×C4×C257—22no—
67C2×C4×C257C2×D411yescharacteristic
68C2×C4×C257C2×D411yescharacteristic
69C2×C4×C257C2×D411yescharacteristic
70C8×C257—41no—
71C8×C257—41no—
72C22×C257#12011yescharacteristic
73C2×D257—22no—
74C2×D257—41no—
75C2×D257—41no—
76C4×C257—21no—
77C4×C257—42no—
78C4×C257C2×SD1611yescharacteristic
79C4×C257C2×Q1611yescharacteristic
80C4×C257AGL(1,ℤ/8)11yescharacteristic
81C4×C257#13711yescharacteristic
82Dic257—22no—
83Dic257—41no—
84Dic257—41no—
85Dic257—81no—
86C2×C257#45011yescharacteristic
87C2×C257#45111yescharacteristic
88C2×C257#45411yescharacteristic
89D257—42no—
90C257#163711yescharacteristic, complemented, Sylow
91#1637—2571nomaximal, Sylow
92D4×Q8—2571no—
93#487—2571no—
94#477—2571no—
95#476—2571no—
96#332D25711yescharacteristic, complemented
97Q8⋊C8—2571no—
98D4⋊C8—2571no—
99C22×Q8—5141no—
100C4⋊Q8—2571no—
101C4⋊Q8—5141no—
102C4⋊Q8C2×D25711yescharacteristic
103C22⋊Q8—5141no—
104C22⋊Q8—5142no—
105C4×Q8—2571no—
106C4×D4—2571no—
107C4×D4—5141no—
108#107—5141no—
109#106—5141no—
110C4⋊C8C2×D25711yescharacteristic
111C4⋊C8C2×D25711yescharacteristic
112#103—5141no—
113#103—5142no—
114#102—5141no—
115C4×C8—2571no—
116C2×Q8—2571no—
117C2×Q8—21no—
118C2×Q8—5142no—
119C2×Q8—5141no—
120C2×Q8—10282no—
121C2×Q8—10282no—
122C2×D4—2571no—
123C22×C4—5142no—
124C22×C4—5141no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—5141no—
128C4⋊C4—2571no—
129C4⋊C4—2571no—
130C4⋊C4—21no—
131C4⋊C4—5142no—
132C4⋊C4—5141no—
133C4⋊C4—5141no—
134C4⋊C4—5141no—
135C4⋊C4D102812yes—
136C22⋊C4—5142no—
137C22⋊C4—5141no—
138C42—5141no—
139C42C22×D25711yescharacteristic
140C23—5141no—
141Q8—2572no—
142Q8—5141no—
143Q8—41no—
144Q8—41no—
145Q8—10282no—
146Q8—10284no—
147D4—2572no—
148D4—5141no—
149C2×C4—22no—
150C2×C4—5141no—
151C2×C4—5141no—
152C2×C4—5141no—
153C2×C4—10284no—
154C2×C4—10281no—
155C2×C4—10281no—
156C2×C4—10281no—
157C2×C4D4×D25711yescharacteristic
158C2×C4D4×D25711yescharacteristic
159C2×C4C2×D102811yescharacteristic
160C8—41no—
161C8—41no—
162C8—10281no—
163C22—5142no—
164C22—10281no—
165C22—10281no—
166C22#2255156211yescharacteristic
167C4—21no—
168C4—5142no—
169C4—42no—
170C4—10281no—
171C4—10281no—
172C4—20561no—
173C4SD16×D25711yescharacteristic
174C4Q16×D25711yescharacteristic
175C4#2255161611yescharacteristic
176C4#2255162111yescharacteristic
177C2—10282no—
178C2#3065688911yescharacteristic
179C2#3065689411yescharacteristic
180C2#3065691211yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#332×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#332D257
hypocenterC257#1637
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #30657089C2semidirect
3. #30657033C2semidirect
5. #30656467C2semidirect
7. #332×C257C2semidirect
90. C257#1637semidirect
96. #332D257semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#306574022
#306570892
#306570332
#306570282
#306564672
#306564642
#332×C2572
#1637257